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Theorem tppreqb 4157
Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)
Assertion
Ref Expression
tppreqb  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  { A ,  B ,  C }  =  { A ,  B } )

Proof of Theorem tppreqb
StepHypRef Expression
1 3ianor 988 . . . 4  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
2 df-3or 972 . . . 4  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
)  <->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  \/  -.  C  =/=  B ) )
31, 2bitri 249 . . 3  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  \/  -.  C  =/=  B ) )
4 orass 522 . . . . 5  |-  ( ( ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  \/ 
-.  C  e.  _V ) 
<->  ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  ( -.  C  =/=  B  \/  -.  C  e.  _V ) ) )
5 ianor 486 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  A ) )
6 tpprceq3 4156 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  A )  ->  { B ,  A ,  C }  =  { B ,  A } )
75, 6sylbir 213 . . . . . . 7  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  ->  { B ,  A ,  C }  =  { B ,  A } )
8 tpcoma 4112 . . . . . . 7  |-  { B ,  A ,  C }  =  { A ,  B ,  C }
9 prcom 4094 . . . . . . 7  |-  { B ,  A }  =  { A ,  B }
107, 8, 93eqtr3g 2518 . . . . . 6  |-  ( ( -.  C  e.  _V  \/  -.  C  =/=  A
)  ->  { A ,  B ,  C }  =  { A ,  B } )
11 orcom 385 . . . . . . . 8  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V ) 
<->  ( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
12 ianor 486 . . . . . . . 8  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  <-> 
( -.  C  e. 
_V  \/  -.  C  =/=  B ) )
1311, 12bitr4i 252 . . . . . . 7  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V ) 
<->  -.  ( C  e. 
_V  /\  C  =/=  B ) )
14 tpprceq3 4156 . . . . . . 7  |-  ( -.  ( C  e.  _V  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B } )
1513, 14sylbi 195 . . . . . 6  |-  ( ( -.  C  =/=  B  \/  -.  C  e.  _V )  ->  { A ,  B ,  C }  =  { A ,  B } )
1610, 15jaoi 377 . . . . 5  |-  ( ( ( -.  C  e. 
_V  \/  -.  C  =/=  A )  \/  ( -.  C  =/=  B  \/  -.  C  e.  _V ) )  ->  { A ,  B ,  C }  =  { A ,  B } )
174, 16sylbi 195 . . . 4  |-  ( ( ( ( -.  C  e.  _V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  \/ 
-.  C  e.  _V )  ->  { A ,  B ,  C }  =  { A ,  B } )
1817orcs 392 . . 3  |-  ( ( ( -.  C  e. 
_V  \/  -.  C  =/=  A )  \/  -.  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B }
)
193, 18sylbi 195 . 2  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  ->  { A ,  B ,  C }  =  { A ,  B }
)
20 df-tp 4021 . . . 4  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
2120eqeq1i 2461 . . 3  |-  ( { A ,  B ,  C }  =  { A ,  B }  <->  ( { A ,  B }  u.  { C } )  =  { A ,  B }
)
22 ssequn2 3663 . . . 4  |-  ( { C }  C_  { A ,  B }  <->  ( { A ,  B }  u.  { C } )  =  { A ,  B } )
23 snssg 4149 . . . . . . 7  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  { C }  C_  { A ,  B } ) )
24 elpri 4036 . . . . . . . 8  |-  ( C  e.  { A ,  B }  ->  ( C  =  A  \/  C  =  B ) )
25 nne 2655 . . . . . . . . . 10  |-  ( -.  C  =/=  A  <->  C  =  A )
26 3mix2 1164 . . . . . . . . . 10  |-  ( -.  C  =/=  A  -> 
( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
2725, 26sylbir 213 . . . . . . . . 9  |-  ( C  =  A  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
28 nne 2655 . . . . . . . . . 10  |-  ( -.  C  =/=  B  <->  C  =  B )
29 3mix3 1165 . . . . . . . . . 10  |-  ( -.  C  =/=  B  -> 
( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
3028, 29sylbir 213 . . . . . . . . 9  |-  ( C  =  B  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3127, 30jaoi 377 . . . . . . . 8  |-  ( ( C  =  A  \/  C  =  B )  ->  ( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) )
3224, 31syl 16 . . . . . . 7  |-  ( C  e.  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3323, 32syl6bir 229 . . . . . 6  |-  ( C  e.  _V  ->  ( { C }  C_  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) ) )
34 3mix1 1163 . . . . . . 7  |-  ( -.  C  e.  _V  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3534a1d 25 . . . . . 6  |-  ( -.  C  e.  _V  ->  ( { C }  C_  { A ,  B }  ->  ( -.  C  e. 
_V  \/  -.  C  =/=  A  \/  -.  C  =/=  B ) ) )
3633, 35pm2.61i 164 . . . . 5  |-  ( { C }  C_  { A ,  B }  ->  ( -.  C  e.  _V  \/  -.  C  =/=  A  \/  -.  C  =/=  B
) )
3736, 1sylibr 212 . . . 4  |-  ( { C }  C_  { A ,  B }  ->  -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B ) )
3822, 37sylbir 213 . . 3  |-  ( ( { A ,  B }  u.  { C } )  =  { A ,  B }  ->  -.  ( C  e. 
_V  /\  C  =/=  A  /\  C  =/=  B
) )
3921, 38sylbi 195 . 2  |-  ( { A ,  B ,  C }  =  { A ,  B }  ->  -.  ( C  e. 
_V  /\  C  =/=  A  /\  C  =/=  B
) )
4019, 39impbii 188 1  |-  ( -.  ( C  e.  _V  /\  C  =/=  A  /\  C  =/=  B )  <->  { A ,  B ,  C }  =  { A ,  B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    u. cun 3459    C_ wss 3461   {csn 4016   {cpr 4018   {ctp 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019  df-tp 4021
This theorem is referenced by: (None)
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